# LMIs in Control/pages/L2 gain of Lure systems

## The System

{\begin{aligned}{\dot {x}}(t)&=Ax(t)+B_{p}p(t)+B_{w}w(t),\\z(t)&=C_{z}x(t)\\p_{i}(t)&=\phi _{i}(q_{i}(t)),i=1,\dots ,n_{p}\\q&=C_{q}x,\\0&\leq \sigma \phi _{i}(\sigma )\leq \sigma ^{2}\ \forall \sigma \in \mathbb {R} \end{aligned}}

## The Data

The matrices $A,B_{p},B_{w},C_{q},C_{z}$ .

## The Optimization Problem:

The following semi-definite problem should be solved.

{\begin{aligned}&\min _{\{P\succ 0,\Lambda =diag(\lambda _{1},\dots ,\lambda _{n_{p}})\succeq 0,T=diag(\tau _{1},\dots ,\tau _{n_{p}})\succeq 0\}}\gamma ^{2}\;\\&\quad \quad \quad \quad \quad \quad \quad \quad s.t.\quad {\begin{bmatrix}A^{\top }P+PA+C_{z}^{\top }C_{z}&PB_{p}+A^{\top }C_{q}^{\top }\Lambda +C_{q}^{\top }T&PB_{w}\\B_{p}^{\top }P+\Lambda C_{q}A+TC_{q}&\Lambda C_{q}B_{p}+B_{p}^{\top }C_{q}^{\top }\Lambda -2T&\Lambda C_{q}B_{w}\\B_{w}^{\top }P&B_{w}^{\top }C_{q}^{\top }\Lambda &-\gamma ^{2}I\end{bmatrix}}\preceq 0\\\end{aligned}}

## Conclusion

The value function returns the square of the smallest provable upper bound on the ${\mathcal {L}}_{2}$  gain of the Lure's system.

## Remark

The Lyapunov function which is used to proof is similar to the one for the systems with unknown parameters.